A two-dimensional space-time absolute nodal coordinates cable element and its application in shape memory alloy

In this paper, a SMA space-time ANCF cable element which based on an asymmetric Brinson constitutive model (Poorasadion model) is proposed. To solve the discontinuity of stress in cross section, 5-order polynomial interpolation in neutral axis is used in shape function to construct the displacement formulation of the presented element. The continuity of position and velocity is also guaranteed by 3-order polynomial interpolation in the time direction. The stress equation based on the stress distribution in the neutral axis and cross section is derived. The phase transition regions are given in detail, and the update strategy of initial state parameter for the constitutive law is determined. In the space-time discretization, a solver based on Hamilton's law of varying action and P2 method is established. As for constrains, the application methods of replaceable constraint and supplementary constraint are proposed to solve the singular problems in Jacobian matrix. Numerical simulation verifies that this method is capable of simulating the cyclic loading of thermo-mechanic problems. A free pendulum simulation shows the difference between the variation of stress and component of detwinned martensite. Simulation results also show that this element could be used for capturing the shape memory effect and pseudo-elasticity effect by SMA cable and the asymmetric effect in tension and compression by adopting an asymmetric constitutive law.